Question

The line segment joining (5, 0) and (10 cos 0, 10 sin $\theta$ ) is divided internally in the ratio 2:3 at $P$ .If $\theta$ varies, then the locus of $P$ is

• A. a pair of straight lines
• B. a circle
• C. a straight line
• D. a parabola
• E. an ellipse

Answer 1

Answer: (B)

Let the coordinates of point $P$ be $(x,y)$. Then,
$\Rightarrow X=\frac{20\cos \theta +15}{5}$
and $y=\frac{20\sin \theta +0}{5}$
$\Rightarrow x=4\cos \theta +3$ and $y=4\sin \theta$
$\Rightarrow x-3=4\cos \theta$ and $y=4\sin \theta$
$\Rightarrow (x-3{)}^2+y^2=16$
which is an equation of a circle.